Question: How can I obtain the acceleration vector in ECI for a fixed observer on Earth with rotation around the Z axis if I know:
$\Omega=7.25*10^{-5}$ rad/s (rotation rate),
position in ECI ($r_{ECI}$)
velocity in ECI ($v_{ECI}$)
and Acceleration in ECEF ($a_{ECEF}$)?
I know that to convert from a rotating observer to a fixed observer the equations are the following (taken from [1]):
$$a_x=\frac{d^2x'}{dt^2}-x'\Omega^2 - y'\frac{d\Omega}{dt}-2\Omega \frac{dy'}{dt}$$
Where the ' superscript denotes ECEF parameters and without it ECI parameters.
But this seems strange to me. In my case I am trying to find the acceleration on an observer on a unit sphere of radius 1, with observer on the point at time zero of
$$r_0=[.25 ;.85 ;.463680924]$$
This vector is valid for both the ECI and ECEF frame, because at time 0 I consider them to be perfectly aligned with each other. After, the ECI frame rotates in a clockwise direction about the z axis.
I know the acceleration in the ECI frame should have some oscillatory behaviour, and in ECEF it should be zero because the observer is not moving relative to te sphere. This is because the first, third and last components cancel out; since the observer has no acceleration in ECEF, nor velocity, and the rotation rate is constant.
$$\frac{d^2x'}{dt}=0$$
$$\frac{d\Omega}{dt}=0$$
$$\frac{dy'}{dt}=0$$
$$a_x=-x'\Omega^2$$
Using the above equation will not give an oscillatory acceleration, which is why I am not sure if its the right one.
References